 

Mirela Jukic Bokun and
Ivan Soldo
Extensions of D(1)pairs in some imaginary quadratic fields
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Published: 
June 3, 2024. 
Keywords: 
Pellian equation, Diophantine mtuple, quadratic field. 
Subject [2020]: 
11D09, 11D45, 11R11. 


Abstract
In this paper, we discuss the extensibility of Diophantine D(1) pairs
{a, b}, where a,b are positive integers in the ring Z[\sqrt{k}], k>0. We prove that families of such D(1)pairs with b=p^{i}q^{j}, where p,q are different odd primes and i,j are positive integers, cannot be extended to quadruples in certain rings Z[\sqrt{k}], where k depends on p^{i},q^{i} and a. Further, we present the result on nonexistence of D(1)quintuples of a specific form in certain imaginary quadratic rings.


Acknowledgements
Authors were supported by the Croatian Science Foundation under the project no. IP2022105008.


Author information
Mirela Jukic Bokun
School of Applied Mathematics and Informatics
University of Osijek
Trg Ljudevita Gaja 6
HR31000 Osijek, Croatia
mirela@mathos.hr
Ivan Soldo
School of Applied Mathematics and Informatics
University of Osijek
Trg Ljudevita Gaja 6
HR31000 Osijek, Croatia
isoldo@mathos.hr

